Assignment III - PART B - Answers

 

PART B 

1. Steps Involved in Hypothesis Testing:

There are typically six steps involved in hypothesis testing:

1. State the Hypotheses:

   • Null Hypothesis (H₀): This is the default statement, often assuming no difference or effect.
   • Alternative Hypothesis (H₁): This is the opposite of the null hypothesis, proposing a difference or effect.

2. Set the Significance Level (α): This is the probability of rejecting the null hypothesis when it's actually true (type I error). Common choices are 0.05 (5%) or 0.01 (1%).

3. Select the Test Statistic: This is a statistical measure used to assess the evidence against the null hypothesis. The choice depends on factors like sample size and data type.

4. Determine the Decision Rule: Based on the significance level and test statistic, you define a critical region (rejection zone). If the test statistic falls within this region, you reject the null hypothesis.

5. Collect and Analyze Data: You gather your sample data and calculate the test statistic.

6. Interpret the Results: Based on the decision rule, you either reject the null hypothesis (evidence suggests a difference) or fail to reject it (insufficient evidence for a difference).

2. Hypothesis Testing Example:

Given: Population Mean (μ) = 0.700, Sample Mean (x̄) = 0.742, Sample Standard Deviation (s) = 0.040, Sample Size (n) = 10.

To test the null hypothesis (H₀: μ = 0.700), we could use a one-sample Z-test. However, for 3 marks, you can simply state the approach:

• Calculate the test statistic (e.g., Z-score).
• Determine the critical value for your chosen significance level (α).
• Compare the test statistic to the critical value.
• If the test statistic falls outside the critical region (far from μ), reject H₀. Otherwise, fail to reject H₀.

Work out :

Testing the Null Hypothesis (One-Sample Z-Test)

We can use a one-sample Z-test to evaluate the null hypothesis (H₀: μ = 0.700) for the population mean, where μ is the actual population mean. Here's the step-by-step solution:

1. Calculate the Z-score:

   The Z-score measures how many standard deviations the sample mean (x̄) deviates from the hypothesized population mean (μ).

   Z = (x̄ - μ) / (s / √n)

   where:

   • x̄ = Sample mean (0.742)
   • μ = Hypothesized population mean (0.700)
   • s = Sample standard deviation (0.040)
   • n = Sample size (10)

   Z = (0.742 - 0.700) / (0.040 / √10) Z ≈ 1.05

2. Choose the Significance Level (α):

   Let's assume a common significance level of α = 0.05 (5%).

3. Determine the Critical Z-value:

   For a two-tailed test (alternative hypothesis: not equal to), we need to consider both positive and negative critical values at α/2 = 0.025 (2.5%) on either tail of the standard normal distribution.

   You can use a Z-table or software to find the critical Z-values. In this case, the critical Z-values are approximately ±1.96.

4. Decision Rule:

   • Reject H₀: If the calculated Z-score falls outside the critical region (<-1.96 or >1.96).
   • Fail to Reject H₀: If the Z-score falls within the critical region (-1.96 to 1.96).

5. Interpretation:

   Our calculated Z-score (1.05) falls within the critical region (-1.96 to 1.96). Based on the decision rule, we fail to reject the null hypothesis (H₀: μ = 0.700) at the 5% significance level.

Conclusion:

With the available information, we don't have sufficient evidence to reject the claim that the population mean is 0.700. However, it's important to consider limitations:

• A sample size of 10 is relatively small. Larger samples might provide more conclusive results.
• We haven't explored the p-value, which gives the exact probability of observing a Z-score as extreme or more extreme than the calculated value, assuming the null hypothesis is true.

For a more comprehensive analysis, consider using statistical software to calculate the p-value and explore the impact of different significance levels.

3. Completely Randomized Design (CRD):

A CRD is an experimental design where treatments are randomly assigned to experimental units. This randomization helps control for extraneous variables and ensures unbiased estimates of treatment effects.

Key features:

• Treatments are randomly assigned to subjects or groups.
• Each subject or group has an equal chance of receiving any treatment.
• Subjects or groups are independent of each other.

4. Confidence Interval for Standard Deviation:

Absolutely, here's how to find the 95% confidence limits for the standard deviation of the electric bulb lights, given a sample standard deviation of 100 hours and a sample size of 200:

1. Chi-Square Distribution: We'll use the chi-square (χ²) distribution to find the confidence limits. This distribution is used for estimating population standard deviation based on sample standard deviation.

2. Confidence Level and Degrees of Freedom:

   • We want a 95% confidence level. This translates to a 2.5% area in each tail of the chi-square distribution (since 100% - 95% = 5%, and half goes in each tail).

   • Degrees of freedom (df) for this problem are n-1, where n is the sample size. df = 200 (sample size) - 1 = 199

3. Look Up Chi-Square Values:

   • We need two chi-square values:

     • Lower chi-square value (χ²lower) with 199 degrees of freedom and a cumulative area of 0.025 (2.5% in the lower tail).
     • Upper chi-square value (χ²upper) with 199 degrees of freedom and a cumulative area of 0.975 (95% + 2.5% in the lower tail).

   • You can find these values using a chi-square table or statistical software.

4. Confidence Limits Formula:

   • The confidence limits for the population standard deviation (σ) are:

     Lower Limit (LL) = s * sqrt( χ²lower / df ) Upper Limit (UL) = s * sqrt( χ²upper / df )

   where:

   • s is the sample standard deviation (100 hours in this case).
   • χ²lower and χ²upper are the chi-square values found in step 3.
   • df is the degrees of freedom (199).

5. Calculate the Confidence Limits:

   • Once you have the chi-square values, plug them into the formula along with s and df.
   • The calculation will give you the lower and upper confidence limits for the population standard deviation of the electric bulb lifetimes.

Note: You'll need to refer to a chi-square table or software to find the specific chi-square values for the given degrees of freedom and cumulative areas.

5. Properties of Estimators:

An estimator is a statistic used to estimate an unknown population parameter. Here are some important properties:

• Unbiasedness: An unbiased estimator, on average, estimates the true parameter accurately. The average of many estimates should be close to the parameter. (Example: Sample mean is an unbiased estimator of population mean)
• Consistency: As the sample size increases, a consistent estimator gets closer and closer to the true parameter value. (Example: Sample proportion is a consistent estimator of population proportion)
• Efficiency: Among unbiased estimators, the efficient one has the smallest variance (less spread around the true value). (Example: Sample mean is generally more efficient than the median)

6. Hypothesis Testing Concepts:

Hypothesis testing is a statistical method to assess claims about a population parameter. Here's a brief explanation:

• We formulate two competing hypotheses: null hypothesis (H₀) stating no effect or difference, and alternative hypothesis (H₁) proposing an effect or difference.
• We choose a significance level (α) to control the risk of wrongly rejecting a true null hypothesis.
• We select a test statistic based on the data and hypotheses.
• We define a decision rule based on the significance level and test statistic.
• We analyze the data and compare the test statistic to the decision rule.
• Based on the comparison, we either reject the null hypothesis (evidence suggests a difference) or fail to reject it (insufficient evidence).

7. Uses of ANOVA Test:

ANOVA (Analysis of Variance) is a statistical technique used to compare the means of several groups. Here are some common uses:

• Comparing the effectiveness of different treatments or interventions.
• Analyzing the impact of multiple factors on an outcome variable.

8. Random Design - Merits & Demerits:

Merits:

• Reduces bias through random treatment assignment.
• Improves generalizability of results to the population.
• Simple to implement and statistically efficient (for few treatments).

Demerits:

• Limited control over extraneous variables.
• May not account for underlying group differences.
• Doesn't capture treatment interactions with other factors.
• Larger sample size might be needed compared to some designs.

Assignment III - PART A - Probable Answers

PART A (Add your own examples and formula in exam)

Here are some acceptable definitions/concepts for the questions:

1. Bias in Estimation: In statistics, bias refers to the systematic tendency of an estimator to consistently overestimate or underestimate the true value of the parameter it's trying to represent. Imagine you're trying to estimate the average height of people in your city. If your sample only includes people from a basketball team, the average height will likely be biased upwards because it doesn't represent the entire population.

2. Mean Squared Error (MSE): This term measures the average squared difference between the estimated values and the actual values. A lower MSE indicates that the estimator, on average, produces predictions closer to the true values. Think of it like an average squared distance between your "guesses" and the bullseye in archery.

3. Two-way ANOVA (Analysis of Variance): This statistical test helps us understand how two categorical factors simultaneously affect a continuous outcome variable. It analyzes the variance (spread) in the data to determine if the differences in means between groups are due to chance or if there's a statistically significant effect from one or both factors, or their interaction. Imagine studying the effect of fertilizer type (A) and watering frequency (B) on plant growth (outcome). A two-way ANOVA would tell you if either factor alone, or both together, significantly influence plant growth.

4. Confidence Intervals: These are ranges of values that are likely to contain the true population parameter with a specific level of confidence (often 95% or 99%). It's like estimating a target's location on a dartboard by throwing multiple darts. The confidence interval is the area where you're reasonably certain the bullseye lies, based on the spread of your throws.

5. Bias vs. Precision:

   • Bias: As mentioned earlier, bias is the systematic error that causes an estimator to consistently deviate from the true value. It reflects how accurate your estimates are on average.
   • Precision: This refers to how close repeated measurements are to each other. It indicates the level of random variation in your estimates. Think of it like the tightness of your dart throws around the target. High precision means your throws are clustered together, even if they're not necessarily hitting the bullseye (due to bias).

6. Parameter: A parameter is a numerical characteristic that describes a population. It's a fixed but unknown value we're trying to estimate. For example, the population mean (average height of all people in your city) is a parameter.

7. One-way ANOVA: This is a statistical test used to compare the means of several groups (more than two) to see if there's a statistically significant difference between their averages. It assumes there's only one factor influencing the outcome variable. Imagine comparing the average exam scores of students taught by different teachers. A one-way ANOVA would tell you if there's a significant difference in teaching effectiveness based on average scores.

8. Sample Size: This refers to the number of observations or data points included in your statistical analysis. A larger sample size generally leads to more reliable estimates and more accurate conclusions from your analysis. It's like having more dart throws; the more throws you have, the better you can estimate the target's location.